と is a quotation particle, and the part following 決意が薄まらないうちに is omitted there since it's obvious from the context.
決意が薄まらないうちに店を出ようとして/考えて、足早に店を出る僕の後ろで
In the expression AないうちにB, when B is a volitional action/an action for a purpose like your example, the meaning is almost the same whether と is put after うちに or not.
On the other hand, when B is a non-volitional or an undesired action, と can't be used.
e.g.
×夜が明けないうちにと、雪が降り始めた。
(雪が降り始める is non-volitional, so 夜が明けないうちに雪が降り始めようとして is ungrammatical.)
乗客が全て乗らないうちに、船は出港してしまった。
船は出港してしまった is an accident/undesired action here.
乗客が全て乗らないうちにと can be possible in a specific context. It means that the ship/captain intentionally did so, for instance, they thought it's dangerous if all the passengers got on the ship.